Generalized linear array model

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In statistics, the generalized linear array model(GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product.

Overview[edit]

The generalized linear array model or GLAM was introduced in 2006.[1] Such models provide a structure and a computational procedure for fitting generalized linear models or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.

Suppose that the data \mathbf Y is arranged in a d-dimensional array with size n_1\times n_2\times\ldots\times n_d; thus,the corresponding data vector \mathbf y = \textbf{vec}(\mathbf Y) has size n_1n_2n_3\cdots n_d. Suppose also that the design matrix is of the form

\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\ldots\otimes\mathbf X_1.

The standard analysis of a GLM with data vector \mathbf y and design matrix \mathbf X proceeds by repeated evaluation of the scoring algorithm

 \mathbf X'\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\mathbf z} ,

where \tilde{\boldsymbol\theta} represents the approximate solution of \boldsymbol\theta, and \hat{\boldsymbol\theta} is the improved value of it; \mathbf W_\delta is the diagonal weight matrix with elements

 w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\text{var}(y_i),

and

\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)

is the working variable.

Computationally, GLAM provides array algorithms to calculate the linear predictor,

 \boldsymbol\eta = \mathbf X \boldsymbol\theta

and the weighted inner product

 \mathbf X'\tilde{\mathbf W}_\delta\mathbf X

without evaluation of the model matrix  \mathbf X .

Example[edit]

In 2 dimensions, let \mathbf X = \mathbf X_2\otimes\mathbf X_1, then the linear predictor is written \mathbf X_1 \boldsymbol\Theta \mathbf X_2' where \boldsymbol\Theta is the matrix of coefficients; the weighted inner product is obtained from G(\mathbf X_1)' \mathbf W G(\mathbf X_2) and  \mathbf W is the matrix of weights; here G(\mathbf M) is the row tensor function of the  r \times c matrix  \mathbf M given by

G(\mathbf M) = (\mathbf M \otimes \mathbf 1') * (\mathbf 1' \otimes \mathbf M)

where * means element by element multiplication and \mathbf 1 is a vector of 1's of length  c.

These low storage high speed formulae extend to d-dimensions.

Applications[edit]

GLAM is designed to be used in d-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of d one-dimensional smoothing matrices.

References[edit]

  1. ^ Currie, I.D.;Durban, M.;Eilers, P. H. C. (2006) "Generalized linear array models with applications to multidimensional smoothing",Journal of the Royal Statistical Society, 68(2), 259-280.